Classical credit risk and pricing models typically assume that the expected recovery at default is constant, or at the very least independent of the default probability. However, a large body of recent empirical evidence has challenged this assumption and shown that default rates are in fact negatively correlated with recovery rates \cite{ABRS}. Recently, Moody's Analytics proposed a model in the context of credit capital which incorporates this empirically observed correlation within a structural framework \cite{LH}. In this work we revisit Moody's PD-LGD correlation model and in the process complete and extend several results. We then price Bond and Credit Default Swaps with recovery risk using the PD-LGD model under both the Merton and Black-Cox default assumptions, and in addition compute associated risk metrics and Greeks. Our results are then compared with classical results which assume no recovery risk.